# How to convert interactions in a classification and regression tree when translating it into a regression model?

CARTs naturally consider interactions between covariates, see:

Do CART trees capture interactions among predictors?

I would like to know whether it is always possible to translate the variable interactions when converting a classification and regression tree into a generalised linear model.

For example, the interaction between economic conditions ($X_1$) and type of building purchased ($X_2$) can be given by a regression specified with Wilkinson notation:

$$Y \sim X_1:X_2$$

That is, find the contribution to the response based on all possible values of the cartesian product of $X_1$ and $X_2$.

Is there a general rule that can be applied to translate an arbitrary interaction in a CART?